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Practical limitations on robustness and scalability of quantum Internet

Abhishek Sadhu, Meghana Ayyala Somayajula, Karol Horodecki, Siddhartha Das

TL;DR

The paper investigates the practical limits on the robustness and scalability of a quantum Internet by embedding quantum-network tasks into a graph-theoretic framework and applying percolation theory. It shows that even with repeater-assisted schemes, long-distance DI-QKD and entanglement distribution face finite-diameter constraints, with end-to-end secret-key distillation becoming impossible when end-to-end visibility falls below critical thresholds such as $\gamma_{crit}^{\theta} \approx 0.7445$ for CHSH-based protocols. It introduces a suite of graph-based metrics (e.g., $\Upsilon$, $\zeta$, $\Gamma$, $\widetilde{w}_{\ast}$, $\nu_i$) to compare topologies and identify critical nodes, and derives time–length trade-offs ($\alpha l_{cr}+\beta t_{cr}$) and fibre-advantage factors $f$ for repeater networks. The work applies these ideas to real-world contexts, including satellite-based city-to-city entanglement distribution, and analyzes current quantum processor networks to benchmark robustness, ultimately providing a roadmap for task-oriented quantum-network design and evaluation under realistic imperfections.

Abstract

As quantum theory allows for information processing and computing tasks that otherwise are not possible with classical systems, there is a need and use of quantum Internet beyond existing network systems. At the same time, the realization of a desirably functional quantum Internet is hindered by fundamental and practical challenges such as high loss during transmission of quantum systems, decoherence due to interaction with the environment, fragility of quantum states, etc. We study the implications of these constraints by analyzing the limitations on the scaling and robustness of quantum Internet. Considering quantum networks, we present practical bottlenecks for secure communication, delegated computing, and resource distribution among end nodes. Motivated by the power of abstraction in graph theory (in association with quantum information theory), we consider graph-theoretic quantifiers to assess network robustness and provide critical values of communication lines for viable communication over quantum Internet. In particular, we begin by discussing limitations on usefulness of isotropic states as device-independent quantum key repeaters which otherwise could be useful for device-independent quantum key distribution. We consider some quantum networks of practical interest, ranging from satellite-based networks connecting far-off spatial locations to currently available quantum processor architectures within computers, and analyze their robustness to perform quantum information processing tasks. Some of these tasks form primitives for delegated quantum computing, e.g., entanglement distribution and quantum teleportation. For some examples of quantum networks, we present algorithms to perform different quantum network tasks of interest such as constructing the network structure, finding the shortest path between a pair of end nodes, and optimizing the flow of resources at a node.

Practical limitations on robustness and scalability of quantum Internet

TL;DR

The paper investigates the practical limits on the robustness and scalability of a quantum Internet by embedding quantum-network tasks into a graph-theoretic framework and applying percolation theory. It shows that even with repeater-assisted schemes, long-distance DI-QKD and entanglement distribution face finite-diameter constraints, with end-to-end secret-key distillation becoming impossible when end-to-end visibility falls below critical thresholds such as for CHSH-based protocols. It introduces a suite of graph-based metrics (e.g., , , , , ) to compare topologies and identify critical nodes, and derives time–length trade-offs () and fibre-advantage factors for repeater networks. The work applies these ideas to real-world contexts, including satellite-based city-to-city entanglement distribution, and analyzes current quantum processor networks to benchmark robustness, ultimately providing a roadmap for task-oriented quantum-network design and evaluation under realistic imperfections.

Abstract

As quantum theory allows for information processing and computing tasks that otherwise are not possible with classical systems, there is a need and use of quantum Internet beyond existing network systems. At the same time, the realization of a desirably functional quantum Internet is hindered by fundamental and practical challenges such as high loss during transmission of quantum systems, decoherence due to interaction with the environment, fragility of quantum states, etc. We study the implications of these constraints by analyzing the limitations on the scaling and robustness of quantum Internet. Considering quantum networks, we present practical bottlenecks for secure communication, delegated computing, and resource distribution among end nodes. Motivated by the power of abstraction in graph theory (in association with quantum information theory), we consider graph-theoretic quantifiers to assess network robustness and provide critical values of communication lines for viable communication over quantum Internet. In particular, we begin by discussing limitations on usefulness of isotropic states as device-independent quantum key repeaters which otherwise could be useful for device-independent quantum key distribution. We consider some quantum networks of practical interest, ranging from satellite-based networks connecting far-off spatial locations to currently available quantum processor architectures within computers, and analyze their robustness to perform quantum information processing tasks. Some of these tasks form primitives for delegated quantum computing, e.g., entanglement distribution and quantum teleportation. For some examples of quantum networks, we present algorithms to perform different quantum network tasks of interest such as constructing the network structure, finding the shortest path between a pair of end nodes, and optimizing the flow of resources at a node.
Paper Structure (37 sections, 6 theorems, 91 equations, 36 figures, 2 tables, 5 algorithms)

This paper contains 37 sections, 6 theorems, 91 equations, 36 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Motivation and our contribution
  3. Summary of the main results
  4. Structure of the paper
  5. Limitations on the network architecture with repeaters
  6. Graph theoretic analysis of networks
  7. Graph theoretic framework of networks
  8. Limitations on repeater networks
  9. Critical success probability for repeater networks
  10. Critical time and length scales for repeater networks
  11. Limitations on quantum network topologies
  12. Robustness measure for networks
  13. Comparison of the robustness of network topologies
  14. Critical nodes in a network
  15. Real world instances
  16. ...and 22 more sections

Key Result

Proposition 1

Consider a standard linear DI key repeater chain with $n$ relay (intermediate) stations between two end nodes. The successive nodes $v_i$ and $v_j$ of the repeater chain share a two-qubit isotropic state $\rho_{ij}^I(p(\lambda^2),2)$ and the relay stations perform standard Bell measurement with succ

Figures (36)

  • Figure 1: A network of point-to-point connections made of isotropic states is depicted. Any two nodes can be made directly connected via entanglement swapping to generate DI-QKD key along connecting path if and only if they are in a finite distance less than a critical one $\leq d_{crit}^{DI-QKD}$. E.g. nodes $u$ and $w$ cannot be connected as they are too far. Only in the yellow area any pair of vertices can be made directly connected in device-independent way. In graph-theoretical language, the diameter of any fully connected subgraph is finite, bounded by $d_{crit}^{DI-QKD}$. The same fact limits distance of connecting nodes with pure entanglement needed for faithful teleportation which enables e.g. delegated quantum computing. (Color online)
  • Figure 3: In this figure, we present a repeater-based network to share isotropic states between Alice and Bob. The shared state is then used to perform DI-QKD protocols. The blue circles in the figure depict qubits. We assume all the repeater stations are equidistant and identical. (Color online)
  • Figure 4: In this figure, we plot the allowed number of relay stations for performing a DI-QKD protocol with non-zero key rates by Alice and Bob as a function of the isotropic state parameter $\lambda$ for different success probability of standard Bell measurement when the critical threshold from Eq. \ref{['eq:critThresholdWerner']} is $\gamma^{\theta}_{\text{crit}} = 0.7445.$ (Color online)
  • Figure 5: A partially connected mesh network with $4$ nodes. In this graph the adjacency matrix $\mathsf{A}$ and the success matrix $\mathbb{f}$ are different. While performing an information processing task using this network say transferring a resource $\chi$ from $v_1 \to{v_2}$, it is preferable to use a cooperative strategy $v_1\to{v_3}\to{v_2}$ over a non-cooperative strategy $v_1\to{v_2}$ as it has a higher success probability. (Color online)
  • Figure 6: In this figure, we present an entanglement swapping-based repeater network. There are sources $S_1 (S_2)$ producing state $\Psi^+$ and sending it to a repeater station and Alice (Bob) via optical fibers of length $l$ (shown in yellow). The qubits are stored in quantum memories at the repeater station and the stations of Alice and Bob for $t$ time steps (shown as self-loops). The repeater station performs standard Bell measurement on its share of qubits. (Color online)
  • ...and 31 more figures

Theorems & Definitions (30)

  • Proposition 1
  • proof
  • Definition 1
  • Definition 2: Critically large network
  • Proposition 2
  • proof
  • Theorem 1
  • Definition 3
  • Example 1
  • Definition 4
  • ...and 20 more